Find the next number in the sequence: 26, 63, 124, 215, 342.....

Concept: A numeral sequence like this is often governed by a hidden closed-form rule that ties each term to its position. Here, every term equals a whole number's cube minus 1 — term = n3 − 1 — where n runs through consecutive integers.

Applying it to this sequence:

1. The five given terms are 26, 63, 124, 215, 342 — check whether each is one less than a perfect cube.

2. 33 = 27, 43 = 64, 53 = 125, 63 = 216, 73 = 343 — these are five consecutive cubes.

3. Subtracting 1 from each cube: 27 − 1 = 26, 64 − 1 = 63, 125 − 1 = 124, 216 − 1 = 215, 343 − 1 = 342 — every given term matches exactly.

4. The next whole number after 7 is 8, so the next term follows the same rule: 83 − 1 = 512 − 1 = 511.

Cross-check: Look at the gaps between consecutive terms: 63−26=37, 124−63=61, 215−124=91, 342−215=127. These gaps themselves grow by 24, 30, and 36 — a steady step of 6 each time. The next step up is 36+6=42, so the next gap is 127+42=169, and adding it to the last term gives 342+169=511, matching the cube-based result.

So the next number in the sequence is 511.